Timeline for Polyhedron - sphere intersection
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 2, 2021 at 10:30 | comment | added | McDuck | And then I want to take a look at the Cauchy's arm lemma (especially the induction part). Although I've been searching for some proofs, but all I found was Cauchy's arm lemma on a growing sphere. | |
| Apr 2, 2021 at 10:27 | comment | added | McDuck | @GevaYashfe I'm looking into Cauchy's rigidity theorem (from Proofs from THE BOOK) and taking it apart. So far I tried to convince myself about the main part (creating a polygon with the vertex with at most 2 changes between black and white in a cyclic order). | |
| Apr 2, 2021 at 10:13 | comment | added | Geva Yashfe | It is much shorter if you are willing to define convex spherical polygons to be finite intersections of closed hemispheres. In this case it follows directly from the fact that a halfspace passing through the center of the sphere intersects the sphere in a closed hemisphere, and a convex polyhedron is a finite intersection of closed halfspaces... I guess this is equivalent to what @JosephO'Rourke wrote above. | |
| Apr 2, 2021 at 10:06 | comment | added | Geva Yashfe | I know no reference, here is an idea for a proof: intersect the sphere with the boundary of the polyhedron (I assume everything lives in $\mathbb{R}^3$). The intersection is the union of sets of the form $S^2 \cap (\text{plane through the center of the sphere})$, each of which is a geodesic on the sphere. You obtain a geodesic spherical polygon. To show convexity look at the angles of this polygon and prove they equal the angles between the corresponding planes. Then look at the intersection of the sphere with the interior of the polyhedron, and show it equals the interior of the polygon. | |
| Apr 2, 2021 at 10:03 | comment | added | Geva Yashfe | @McDuck Can you explain the context, and your definition for a convex polyhedron and a spherical polygon? Is this all in $\mathbb{R}^3$ or is the dimension intended to be general? | |
| Apr 2, 2021 at 8:22 | comment | added | McDuck | @JosephO'Rourke Oh really? Didn't know that, thanks! (Not sarcasm). | |
| Apr 1, 2021 at 23:48 | review | Close votes | |||
| Apr 15, 2021 at 0:53 | |||||
| Apr 1, 2021 at 22:10 | comment | added | Joseph O'Rourke | To me it is unclear what you are asking, because the definition of a convex vertex of a polyhedron is that the intersection with a sufficiently small sphere (centered on the vertex) is convex. | |
| Apr 1, 2021 at 21:51 | review | First posts | |||
| Apr 1, 2021 at 23:33 | |||||
| Apr 1, 2021 at 21:46 | history | asked | McDuck | CC BY-SA 4.0 |